Photonic crystals (PC) are periodic dielectric structures which can prohibit the propagation of light in certain frequency ranges. Photonic crystals have spatially periodic variations in refractive index and with a sufficiently high contrast in refractive index, photonic bandgaps can be opened in the structure's optical spectrum. The “photonic bandgap” is the frequency range within which propagation of light through the photonic crystal is prevented. A photonic crystal that has spatial periodicity in three dimensions can prevent light having a frequency within the crystal's photonic bandgap from propagating in any direction. However, fabrication of such a structure is technically challenging. A more attractive alternative is to utilize photonic crystal slabs that are two-dimensionally periodic dielectric structures of finite height that have a band gap for propagation in the plane and use index-confinement in the third dimension. In addition to being easier to fabricate, two-dimensional photonic crystal slabs provide the advantage that they are compatible with the planar technologies of standard semiconductor processing.
An example of a two-dimensional photonic crystal structure periodic in two dimensions and homogeneous in the third may be fabricated from a bulk material having a periodic lattice of circular air filled columns extending through the bulk material in the height direction and periodic in the planar direction. The propagation of light in two-dimensional photonic crystals is determined by a number of parameters, including radius of the cylindrical columns, the lattice spacing, the symmetry of the lattice and the refractive indices of the bulk and column material.
Introducing defects in the periodic structure of a photonic crystal allows the existence of localized electromagnetic states that are trapped at the defect site and that have resonant frequencies within the bandgap of the surrounding photonic crystal material. By providing a line of such defects in the photonic crystal, a waveguiding structure is created that can be used in the control and guiding of light (see, for example, J. D. Joannopoulos, R. D. Meade, and J. N. Winn, “Photonic Crystals”, Princeton University Press, Princeton, N.J., 1995). Light of a given frequency that is prevented from propagating in the photonic crystal may propagate in the defect region.
A two-dimensional photonic crystal slab waveguide usually comprises a two-dimensional periodic lattice in the form of an array of dielectric rods or air holes incorporated in a slab body. High guiding efficiency can be achieved only in a narrow frequency region close to the upper or lower edge (for dielectric rods or air holes, respectively) of the waveguide band, where there are no leaky modes. Typically, high guiding efficiency is achieved only in a narrow frequency region that is only a few percent of the center frequency of the waveguide band and existing configurations suffer from low group velocities in the allowed waveguide band. Low group velocity increases the unwanted effects of disorder and absorption.(see S. G. Johnson, S. Fan, P. R. Villeneuve, L. Kolodziejski and J. D. Joannopoulos, Phys. Rev. B 60, 5751, 1999 and S. G. Johnson, P. R. Villeneuve, S. Fan and J. D. Joannopoulos, Phys. Rev. B 62, 8212, 2000).
FIG. 1 shows an xy view of prior art two-dimensional photonic crystal slab apparatus 100. Photonic crystal slab 115 has circular holes 110 arranged to from a periodic triangular lattice with a lattice spacing equal to a. Circular holes 110 are filled with air. Region of defects 125 is created by replacing circular holes 110 of the lattice with larger circular holes 120 along a line in the x direction. Ridge waveguide 175 couples light into photonic crystal slab apparatus 100 that may have its edge at line A′, line B′ or line C′ in FIG. 1.
FIG. 2 shows the transmission coefficient for two-dimensional crystal slab apparatus 100 as a function of frequency expressed in fractions of c/a where cis the speed of light and a is the lattice spacing. The radius for circular holes 120 is about 0.45a and the radius for circular holes 110 is about 0.3a. Curve 210 represents the unguided case which has low transmission in the bandgap and high transmission in the allow band. Curve 201 represents the case where ridge waveguide 175 is attached to photonic crystal slab 115 at the edge defined by line A in FIG. 1. Curve 202 represents the case where ridge waveguide 175 is connected to photonic crystal slab 115 at the edge defined by line B in FIG. 1. Curve 203 represents the case where ridge waveguide 175 is connected to photonic crystal slab 115 at the edge defined by line C′ in FIG. 2. The transmission for curve 203 is a maximum for a frequency of about 0.253c/a and the waveguide band is narrow. Increasing the radius of circular holes 120 to 0.5a causes circular holes 120 to touch and start to overlap. This results in rapid deterioration of the transmission properties of two-dimensional crystal slab apparatus 100 as the light wave becomes less confined due to the decrease of the average dielectric constant of two-dimensional crystal slab 100.